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Question

Every student at Darcy School is in at least one of three clubs: horseback riding, embroidery, and country dancing, which are the only clubs in existence at the school. The ratio of the number of students in exactly two clubs to the number of students in exactly one club is 4:3, while the ratio of the number of students in exactly two clubs to the number of students in at least two clubs is 5:7. Which of the following could be the total number of students at Darcy School?

A. 63
B. 69
C. 74
D. 82
E. 86

Answer

Although the setup of this problem seems to call for a Venn Diagram with three circles, this question is actually solely about ratios. We need to figure out the ratio of students who are in only one club to the ratio of students in exactly two clubs to the ratio of students in all three clubs.

One : Two : Three Clubs = ?

We can use the unknown multiplier to help us represent the two ratios. The ratio of the number of students in exactly two clubs to the number of students in exactly one club is 4:3.

And the ratio of the number of students in exactly two clubs to the number of students in at least two clubs is 5:7. Make sure to use a different variable for this ratio. Also, note that this ratio is students in 2 clubs to students in at least two clubs. That means that there are really only 2 students in three clubs for every 5 people in two clubs.

Now we have two different ways of expressing the number of students in exactly two clubs. We know that the number of students in two clubs is a multiple of both 4 and 5. Therefore, the number of students in two clubs must be a multiple of 20. Multiply the first ratio by 5 and the second ratio by 4.

We can now combine the two ratios into one ratio, which will be 15z : 20z : 8z.

The number of students in the school must be a multiple of 15 + 20 + 8 = 43. The only answer that is a multiple of 43 is 86.

The correct answer is E.

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3 comments

I was close but no cigar. Can you provide another way of explaining how the second ratio is reduced or how to get to its true meaning? Specially, "2 students in three clubs for every 5 people in two clubs?"

same as aaron, i am little confused by how the second ratio is reduced.

Is this thought process correct: Ratio of students in 2 clubs : ratio of students in at least 2 clubs is 5:7 --> out of the 7 students who are a member of at least 2 clubs, 5 students are already members of 2 clubs --> 7-5 = 2 who are members of all the 3 clubs?